For a non-trivial connected graph G, a set S ⊆ V (G) is called an edge geodetic set of G if every edge of G is contained in a geodesic joining some pair of vertices in S. The edge geodetic number g1(G) of G is the minimum order of its edge geodetic sets and any edge geodetic set of order g1(G) is an edge geodetic basis. A connected edge geodetic set of G is an edge geodetic set S such that the ...

Let p be a prime and n a positive integer. In [J. Austral. Math. Soc. 81 (2006), 153–164], Feng and Kwak showed that if p > 5 then every connected cubic symmetric graph of order 2p is a Cayley graph. Clearly, this is not true for p = 5 because the Petersen graph is non-Cayley. But they conjectured that this is true for p = 3. This conjecture is confirmed in this paper. Also, for the case when p...

Given a directed graph G, the K node-disjoint paths problem consists in finding a partition of G into K node-disjoint paths, such that each path ends up in a given subset of nodes in G. This article provides a necessary condition for the K node-disjoint paths problem which combines (1) the structure of the reduced graph associated with G, (2) the structure of each strongly connected component o...

A clustered graph C = (G; T ) consists of an undirected graph G and a rooted tree T in which the leaves of T correspond to the vertices of G = (V;E). Each vertex c in T corresponds to a subset of the vertices of the graph called \cluster". c-planarity is a natural extension of graph planarity for clustered graphs, and plays an important role in automatic graph drawing. The complexity status of ...

In 1981, Duffus, Gould, and Jacobson showed that every connected graph either has a Hamiltonian path, or contains claw (K1,3) net (a fixed six-vertex graph) as an induced subgraph. This implies subject to being connected, these two are the only minimal (under taking subgraphs) graphs with no path. Brousek (1998) characterized 2-connected, non-Hamiltonian do not contain We characterize 2-connect...

We show that every 3-connected claw-free graph which contains no induced copy of P11 is hamiltonian. Since there exist non-hamiltonian 3-connected claw-free graphs without induced copies of P12 this result is, in a way, best possible. 1. Statement of the main result A graph G is {H1, H2, . . . Hk}-free if G contains no induced subgraphs isomorphic to any of the graphs Hi, i = 1, 2, . . . , k. A...

‎The Steiner distance of a graph‎, ‎introduced by Chartrand‎, ‎Oellermann‎, ‎Tian and Zou in 1989‎, ‎is a natural generalization of the‎ ‎concept of classical graph distance‎. ‎For a connected graph $G$ of‎ ‎order at least $2$ and $Ssubseteq V(G)$‎, ‎the Steiner‎ ‎distance $d(S)$ among the vertices of $S$ is the minimum size among‎ ‎all connected subgraphs whose vertex sets contain $S$‎. ‎Let $...

This paper concerns finite, edge-transitive direct and strong products, as well as infinite weak Cartesian products. We prove that the direct product of two connected, non-bipartite graphs is edge-transitive if and only if both factors are edgetransitive and at least one is arc-transitive, or one factor is edge-transitive and the other is a complete graph with loops at each vertex. Also, a stro...

For each rational number q ≥ 1, we describe two partitions of the vertex set of a graph G, called the q-brick partition and the q-superbrick partition. The special cases when q = 1 are the partitions given by the connected components and the 2-edge-connected components of G, respectively. We obtain structural results on these partitions and describe their relationship to the principal partition...

An edge e of a simple 3-connected graph G is essential if neither the deletion G\e nor the contraction G/e is both simple and 3–connected. Tutte’s Wheels Theorem proves that the only simple 3–connected graphs with no non-essential edges are the wheels. In earlier work, as a corollary of a matroid result, the authors determined all simple 3-connected graphs with at most two non-essential edges. ...